论文标题

关于单调三角传输图的表示和学习

On the representation and learning of monotone triangular transport maps

论文作者

Baptista, Ricardo, Marzouk, Youssef, Zahm, Olivier

论文摘要

度量的运输提供了一种多功能方法,用于建模复杂的概率分布,并具有密度估计,贝叶斯推理,生成建模以及更远的应用。单调三角传输地图$ \ unicode {x2014} $近似值$ \ unicode {x2013} $ rosenblatt(kr)重新排列$ \ unicode {x2014} $是这些任务的规范选择。然而,此类地图的表示和参数化对它们的一般性和表现力以及对从数据学习地图学习(例如,通过最大似然估计)中产生的优化问题的属性有重大影响。我们提出了一个通用框架,用于通过平滑函数的可逆变换来表示单调三角图。我们建立了有关转化的条件,以使相关的无限维度最小化问题没有虚假的局部最小值,即所有局部最小值都是全球最小的。我们展示了满足某些尾部条件的目标分布,唯一的全局最小化器与KR地图相对应。鉴于来自目标的样本,我们提出了一种自适应算法,该算法估计了基础KR映射的半参数近似。我们演示了如何将该框架应用于关节和条件密度估计,无似然推理以及有向图形模型的结构学习,并在一系列样本量中具有稳定的概括性能。

Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps$\unicode{x2014}$approximations of the Knothe$\unicode{x2013}$Rosenblatt (KR) rearrangement$\unicode{x2014}$are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.

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